The Combinatorics of the Garsia-Haiman Modules for Hook Shapes

Ron M. Adin, Jeffrey B. Remmel, Yuval Roichman

Abstract

Several bases of the Garsia-Haiman modules for hook shapes are given, as well as combinatorial decomposition rules for these modules. These bases and rules extend the classical ones for the coinvariant algebra of type $A$. We also give a decomposition of the Garsia-Haiman modules into descent representations.